Are the mean and the median the exact same in this distribution? why or why not?

This answer follows the same idea as Glen B, but with some slightly different story and visual examples

The median and the mean are both measures that can be seen as splitting a distribution into two parts that have equal weights on both sides.

For the mean and the median, these weights on both sides are different measures. They consider different absolute partial moments. They are both integrals of the absolute difference $|x-m|$ but with different powers.

Symmetric distributions

For symmetric distributions, these two sides are automatically the same when the split is made in the plane of symmetry.

It works the same for mean as for the median which are integrals of a left side and a right side that become equal if the two sides have the same shape.

So the point of the plane of symmetry is equal to the mean and it is equal to the median. And the median and mean will be equal (but they do not need this symmetry to be equal)

Asymmetric distributions

For asymmetric distribution, we do not need to have automatically that the dividing plane for the median (giving equal weights of probability on both sides) is also the dividing plane for the mean (giving equal weights of average distance on both sides), and vice-versa.

It is also very typical for asymmetric distributions to have unequal mean and variance.

The only counter-example among common asymmetric distributions that comes to my mind is the binomial distribution where $np$ is an integer and $p \neq 0.5$ (a worked-out example is in Nick Cox's answer with $n=5$ and $p=0.2$) such that median and mean are the same while the distribution is asymmetric.

For continuous distributions, I do not know a common distribution that is both asymmetric and has equal mean and median.

However, it is not difficult to construct a counter-example. The only thing that is needed is to transform a distribution and scale the distances on the left and right sides appropriately such that they have both equal mass and also the equal average distance.

Below is a counterexample where we have a hypothetical distribution that is composed of an equal fifty-fifty mixture of two distributions, on the right side a $\chi^2$-distribution and on the left side a Weibull distribution. By selecting parameters of these distributions such that the means are equal, we get that this left and right side have the same weights.

The distribution is obviously asymmetric but both sides have the same absolute 0-th and 1-th partial moment, namely both sides have 50% of the probability mass and the average absolute distance from the center is 5.

In this sense, the question looks a bit similar like, "Is it possible to have distributions with different shapes but with the same mean?".

Unimodal

Edit: I missed the 'unimodal' specification. To get this we can do the same trick and use a mixture distribution. But this time we need to have both sides with the same mode as well. To find this example I took three distributions with each a mean equal to 1 (exponential distribution, half-logistic distribution scaled by $log(4)$, half-normal distribution scaled by $\sqrt{2/\pi}$) and add two of them together in order to get the same peak height.

In mathematics and statistics, the mean or the arithmetic mean of a list of numbers is the sum of the entire list divided by the number of items in the list. When looking at symmetric distributions, the mean is probably the best measure to arrive at central tendency. In probability theory and statistics, a median is that number separating the higher half of a sample, a population, or a probability distribution, from the lower half.

How to calculate

The Mean or average is probably the most commonly used method of describing central tendency. A mean is computed by adding up all the values and dividing that score by the number of values. The arithmetic mean of a sample

The Median is the number found at the exact middle of the set of values. A median can be computed by listing all numbers in ascending order and then locating the number in the center of that distribution. This is applicable to an odd number list; in case of an even number of observations, there is no single middle value, so it is a usual practice to take the mean of the two middle values.

Example

Let us say that there are nine students in a class with the following scores on a test: 2, 4, 5, 7, 8, 10, 12, 13, 83. In this case the average score (or the mean) is the sum of all the scores divided by nine. This works out to 144/9 = 16. Note that even though 16 is the arithmetic average, it is distorted by the unusually high score of 83 compared to other scores. Almost all of the students' scores are below the average. Therefore, in this case the mean is not a good representative of the central tendency of this sample.

The median, on the other hand, is the value which is such that half the scores are above it and half the scores below. So in this example, the median is 8. There are four scores below and four above the value 8. So 8 represents the mid point or the central tendency of the sample.

Comparison of mean, median and mode of two log-normal distributions with different skewness.

Mean is not a robust statistic tool since it cannot be applied to all distributions but is easily the most widely used statistic tool to derive the central tendency. The reason that mean cannot be applied to all distributions is because it gets unduly impacted by values in the sample that are too small to too large.

The disadvantage of median is that it is difficult to handle theoretically. There is no easy mathematical formula to calculate the median.

Other Types of Means

There are many ways to determine the central tendency, or average, of a set of values. The mean discussed above is technically the arithmetic mean, and is the most commonly used statistic for average. There are other types of means:

Geometric Mean

The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x1,x2,...,xn, the geometric mean is defined as

Geometric means are better than arithmetic means for describing proportional growth. For example, a good application for geometric mean is calculating the compounded annual growth rate (CAGR).

Harmonic Mean

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. The harmonic mean H of the positive real numbers x1,x2,...,xn is

A good application for harmonic means is when averaging multiples. For exampe, it is better to use weighted harmonic mean when calculating the average price–earnings ratio (P/E). If P/E ratios are averaged using a weighted arithmetic mean, high data points get unduly greater weights than low data points.

Pythagorean Means

The arithmetic mean, geometric mean and harmonic mean together form a set of means called the Pythagorean means. For any set of numbers, the harmonic mean is always the smallest of all Pythagorean means, and the arithmetic mean is always the largest of the 3 means. i.e. Harmonic mean ≤ Geometric mean ≤ Arithmetic mean.

Other meanings of the words

Mean can be used as a figure of speech and holds a literary reference. It is also used to imply poor or not being great. Median, in a geometric reference, is a straight line passing from a point in the triangle to the centre of the opposite side.

References

• wikipedia:Mean
• wikipedia:Median
• Modes, Medians and Means: A Unifying Perspective
• Pythagorean means

Can the mean and median be the same?

Why is mean and median the same?

Is the mean always equal to the median in a normal distribution?

In what type of distribution are the mean, median and mode the same or equal?